PUB - Publikationen an der Universität Bielefeld

We study the distribution of the largest eigenvalue in formal Hermitianone-matrix models at multicriticality, where the spectral density acquires anextra number of k-1 zeros at the edge. The distributions are directly expressedthrough the norms of orthogonal polynomials on a semi-infinite interval, as analternative to using Fredholm determinants. They satisfy non-linear recurrencerelations which we show form a Lax pair, making contact to the stringliterature in the early 1990's. The technique of pseudo-differential operatorsallows us to give compact expressions for the logarithm of the gap probabilityin terms of the Painleve XXXIV hierarchy. These are the higher order analoguesof the Tracy-Widom distribution which has k=1. Using known Backlundtransformations we show how to simplify earlier equivalent results that arederived from Fredholm determinant theory, valid for even k in terms of thePainleve II hierarchy.